Both Uses

besiege
in
The Aeneid
(Auto-generated)

This little spot of land, which Heav'n bestows, On ev'ry side is hemm'd with warlike foes; Gaetulian cities here are spread around, And fierce Numidians there your frontiers bound; Here lies a barren waste of thirsty land, And there the Syrtes raise the moving sand; Barcaean troops besiege the narrow shore, And from the sea Pygmalion threatens more.†
Book 4

besiege = surround and cut-off supplies; or overwhelm
Of various forms unnumber'd specters more, Centaurs, and double shapes, besiege the door.†
Book 6 *

Definitions:

(1) (besiege)
to surround a place with armed forces and cut off supplies or attack; or more generally, to overwhelm someone with many requests, complaints, or attention, as if surrounding them and not letting up
(2) (meaning too rare to warrant focus)